When Can One Obtain Certificates of Optimality Using Positivstellensaetze?

The paper investigates how to obtain certificates of positivity and optimality for learning problems whose objectives and constraints are not necessarily polynomial. It isolates an axiomatic core of Fischer's constructive strict and weak Positivstellensätze and extends the resulting theorems to abstract function algebras over ordered fields. The framework distinguishes between objective/constraint functions built from broad classes of continuous or definable operations and auxiliary primitives that satisfy explicit scalar and closure axioms, providing instances over continuous and definable function algebras, including fields not closed under square roots, and analyzing lower-bound and global-optimality certificates as well as computational complexity.

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